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**Overview of part 1 Historical perspective**

From reductionism to systems and networks Examples of complex networks Structural/topological metrics Average path length Degree distribution Clustering Topological models Regular, random, small-world, scale-free networks An evolutionary model of network growth: Preferential attachment Implications of scale-free property in: Robustness/fragility Epidemics/diffusion processes Focusing on the “small scale”: network motifs Networks as functioning circuits

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**References I used many slides from other talks:**

S.Maslov, “Statistical physics of complex networks” I.Yanai, “Evolution of networks” D.Bonchev, “Networks basics” Eileen Kraemer, “Topology and dynamics of complex networks”

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**Historical perspective**

In the beginning.. there was REDUCTIONISM All we need to know is the behavior of the system elements Particles in physics, molecules or proteins in biology, communication links in the Internet Complex systems are nothing but the result of many interactions between the system’s elements No new phenomena will emerge when we consider the entire system A centuries-old very flawed scientific tradition..

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**Historical perspective**

During the 80’s and early 90’s, several parallel approaches departed from reductionism Consider the entire SYSTEM attempting to understand/explain its COMPLEXITY B. Mandelbrot and others: Chaos and non-linear dynamical systems (the math of complexity) P. Bak: Self-Organized Criticality – The edge of chaos S. Wolfram: Cellular Automata S. Kauffman: Random Boolean Networks I. Prigogine: Dissipative Structures J. Holland: Emergence H. Maturana, F. Varela: Autopoiesis networks & cognition Systems Biology

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**Historical perspective**

Systems approach: thinking about Networks The focus moves from the elements (network nodes) to their interactions (network links) To a certain degree, the structural details of each element become less important than the network of interactions Some system properties, such as Robustness, Fragility, Modularity, Hierarchy, Evolvability, Redundancy (and others) can be better understood through the Networks approach Some milestones: 1998: Small-World Networks (D.Watts and S.Strogatz) 1999: Scale-Free Networks (R.Albert & A.L.Barabasi) 2002: Network Motifs (U.Alon)

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Some relevant Zen: Things derive their being and nature by mutual dependence and are nothing in themselves. -Nagarjuna, second century Buddhist philosopher An elementary particle is not an independently existing, unanalyzable entity. It is, in essence, a set of relationships that reach outward to other things. -H.P. Stapp, twentieth century physicist See slides by Itay Yanai

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**Overview of part 1 Examples of complex networks Historical perspective**

From reductionism to systems and networks Examples of complex networks Structural/topological metrics Average path length Degree distribution Clustering Topological models Regular, random, small-world, scale-free networks An evolutionary model of network growth: Preferential attachment Implications of scale-free property in: Robustness/fragility Epidemics/diffusion processes Focusing on the “small scale”: network motifs Networks as functioning circuits

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**Air Transportation Network**

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Actors’ web Kraemer

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**Mathematicians & Computer Scientists**

Kraemer

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Sexual contacts: M. E. J. Newman, The structure and function of complex networks, SIAM Review 45, (2003).

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**High school dating: Data drawn from Peter S**

High school dating: Data drawn from Peter S. Bearman, James Moody, and Katherine Stovel visualized by Mark Newman

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**Internet as measured by Hal Burch and Bill Cheswick's Internet Mapping Project.**

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Metabolic networks KEGG database:

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**Transcription regulatory networks**

Single-celled eukaryote: S. cerevisiae Bacterium: E. coli

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**Bio-Map _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ GENOME PROTEOME**

L-A Barabasi GENOME miRNA regulation? _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ PROTEOME - - protein-gene interactions protein-protein interactions Citrate Cycle METABOLISM Bio-chemical reactions

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C. elegans neuronal net

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**Freshwater food web by Neo Martinez and Richard Williams**

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**Overview of part 1 Structural/topological metrics Average path length**

Historical perspective From reductionism to systems and networks Examples of complex networks Structural/topological metrics Average path length Degree distribution Clustering Topological models Regular, random, small-world, scale-free networks An evolutionary model of network growth: Preferential attachment Implications of scale-free property in: Robustness/fragility Epidemics/diffusion processes Focusing on the “small scale”: network motifs Networks as functioning circuits

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Networks As Graphs Networks can be undirected or directed, depending on whether the interaction between two neighboring nodes proceeds in both directions or in only one of them, respectively. 1 2 3 4 5 6 The specificity of network nodes and links can be quantitatively characterized by weights 2.5 7.3 3.3 12.7 8.1 5.4 Vertex-Weighted Edge-Weighted Bonchev

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Networks As Graphs - 2 A network can be connected (presented by a single component) or disconnected (presented by several disjoint components). connected disconnected Networks having no cycles are termed trees. The more cycles the network has, the more complex it is. trees cyclic graphs Bonchev

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**Networks As Graphs - 3 Some Basic Types of Graphs Paths Stars Cycles**

Complete Graphs Bipartite Graphs Bonchev

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**Structural metrics: Average path length**

Slides by Kraemer & Barabasi, Bonabeau (SciAm’03)

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**Structural Metrics: Degree distribution(connectivity)**

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**Structural Metrics: Clustering coefficient**

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**Several other graph metrics exist**

We will study them as needed Centrality Betweenness Assortativity Modularity …

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Network Evolution Slide by Kraemer

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**Overview of part 1 Topological models**

Historical perspective From reductionism to systems and networks Examples of complex networks Structural/topological metrics Average path length Degree distribution Clustering Topological models Regular, random, small-world, scale-free networks An evolutionary model of network growth: Preferential attachment Implications of scale-free property in: Robustness/fragility Epidemics/diffusion processes Focusing on the “small scale”: network motifs Networks as functioning circuits

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Regular networks

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**Regular networks – fully connected**

Slides by Kraemer & Barabasi, Bonabeau (SciAm’03)

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**Regular networks –Lattice**

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**Regular networks –Lattice: ring world**

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Random networks

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**Random networks (Erdos-Renyi, ‘60)**

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Random Networks

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Small-world networks

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**Small-world networks (Watts-Strogatz, ‘98)**

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Small-world networks

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Small-world networks

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Small-world networks

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Scale-free networks

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Scale-free networks

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Scale-free networks

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Scale-free networks

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**Connectivity distributions for metabolic networks**

E. coli (bacterium) A. fulgidus (archaea) averaged over 43 organisms C. elegans (eukaryote) Jeong et al. Nature (2000)

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**Protein-protein interaction networks**

Jeong et al. Nature 411, (2001) Wagner. RSL (2003)

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**Preferential attachment model**

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**A simple model for generating “scale-free” networks**

Evolution : networks expand continuously by the addition of new vertices, and Preferential-attachment (rich get richer) : new vertices attach preferentially to sites that are already well connected. Barabasi & Bonabeau Sci. Am. May Barabasi and Albert. Science (1999)

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**Scale-free network model**

To incorporate the growing character of the network, starting with a small number (m0) of vertices, at every time step we add a new vertex with m (< m0 ) edges that link the new vertex to m different vertices already present in the system. Barabasi and Albert. Science (1999)

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**Scale-free network model**

To incorporate preferential attachment, we assume that the probability P that a new vertex will be connected to vertex i depends on the connectivity k i of that vertex, so that P(k i ) = k i /S j k j . Barabasi and Albert. Science (1999)

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**Scale-free network model**

This network evolves into a scale-invariant state with the probability that a vertex has k edges, following a power law with an exponent = 2.9 +/- 0.1 After t time steps, the model leads to a random network with t + m0 vertices and mt edges. Barabasi and Albert. Science (1999)

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**Overview of part 1 Implications of scale-free property in:**

Historical perspective From reductionism to systems and networks Examples of complex networks Structural/topological metrics Average path length Degree distribution Clustering Topological models Regular, random, small-world, scale-free networks An evolutionary model of network growth: Preferential attachment Implications of scale-free property in: Robustness/fragility Epidemics/diffusion processes Focusing on the “small scale”: network motifs Networks as functioning circuits

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**Robustness/fragility of scale-free networks**

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**Robustness/fragility**

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**Robustness/fragility**

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**Yeast protein-protein interaction networks**

the phenotypic effect of removing the corresponding protein: Lethal Slow-growth Non-lethal Unknown Jeong et al. Nature 411, (2001)

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**Epidemics & other diffusion processes in scale-free networks**

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**Epidemics in complex networks**

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**Node dynamics and self-organization: Epidemics in complex networks**

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**Results can be generalized to generic **

scale-free connectivity distributions P(k)~ k-g If 2 < g 3 we have absence of an epidemic threshold. If 3 < g 4 an epidemic threshold appears, but it is approached with vanishing slope. If g > 4 the usual MF behavior is recovered. SF networks are equal to random graph. Pastor-Satorras & Vespignani (2001, 2002), Boguna, Pastor-Satorras, Vespignani (2003), Dezso & Barabasi (2001), Havlin et al. (2002), Barthélemy, Barrat, Pastor-Satorras, Vespignani (2004)

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**Overview of part 1 Focusing on the “small scale”: network motifs**

Historical perspective From reductionism to systems and networks Examples of complex networks Structural/topological metrics Average path length Degree distribution Clustering Topological models Regular, random, small-world, scale-free networks An evolutionary model of network growth: Preferential attachment Implications of scale-free property in: Robustness/fragility Epidemics/diffusion processes Focusing on the “small scale”: network motifs Networks as functioning circuits

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Reference Uri Alon, “An Introduction to Systems Biology: Design Principles of Biological Circuits”, Chapman & Hall, 2007

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Definition of motifs Network motifs are subgraphs that occur significantly more often in a real network than in the corresponding randomized network.

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Network motifs Random version of original network Original network

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**Motifs in genetic network of yeast**

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**Motifs in genetic network of E. coli**

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**Examples of network motifs (3 nodes)**

X Y Z Feed forward loop Found in many transcriptional regulatory networks X Y Z coherent incoherent

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**Possible functional role of a coherent feed-forward loop**

Noise filtering: short pulses in input do not result in turning on Z

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**Conservation of network motif constituents**

Drosophila melanogaster Homo Sapiens Mus musculus Orthologs Four nodes motif Saccharomyces cerevisiae C. elegans Arabidopsis thaliana

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